This is the first time I ask a question here, so sorry if I make any mistake in the way I ask it. I'm studing Fefferman's article Pointwise Convergence of Fourier Series, and I have two questions:

1) In the first page, it is defined a function $n(\cdot)$. How can I prove that it is measurable?

2) In the third page, I understand the proof of $||T_p||_2\leq CA_0(p)^{1/2}$. The author says that it is easy to prove the opposite inequality. How can I manage to do that?

3) The author considers intervals $I$ or $\omega$ and denotes by $I^*$ and $\omega^*$ the *double* of $I$ and $\omega$ respectively. What is exactly the *double* of an interval?

Thank you for your answers.